The value of $\int \limits \frac {dx}{x^2(x^4+1)^{3/4}}$ is
- $\bigg (\frac {x^4+1}{x^4}\bigg )^\frac {1}{4}+ c $
- $(x^4+1)^\frac {1}{4}+c$
- $-(x^4+1)^\frac {1}{4}+c$
- $- \bigg (\frac {x^4+1}{x^4} \bigg )^ \frac {1}{4}+c$
The Correct Option is D
Solution and Explanation
put $1+ \frac {1}{x^4}=t^4 \Rightarrow \frac {-4}{x^5}dx=4t^3dt \Rightarrow \frac {dx}{x^5}=-t^3dt$
Hence, the integral becomes
$\int \limits \frac {-t^3dt}{t^3}=- \int \limits dt=-t+c=- \bigg (1+ \frac {1}{x^4} \bigg )^{1/4}+ c$
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Concepts Used:
Methods of Integration
Given below is the list of the different methods of integration that are useful in simplifying integration problems:
Integration by Parts:
If f(x) and g(x) are two functions and their product is to be integrated, then the formula to integrate f(x).g(x) using by parts method is:
∫f(x).g(x) dx = f(x) ∫g(x) dx − ∫(f′(x) [ ∫g(x) dx)]dx + C
Here f(x) is the first function and g(x) is the second function.
Method of Integration Using Partial Fractions:
The formula to integrate rational functions of the form f(x)/g(x) is:
∫[f(x)/g(x)]dx = ∫[p(x)/q(x)]dx + ∫[r(x)/s(x)]dx
where
f(x)/g(x) = p(x)/q(x) + r(x)/s(x) and
g(x) = q(x).s(x)
Integration by Substitution Method
Hence the formula for integration using the substitution method becomes:
∫g(f(x)) dx = ∫g(u)/h(u) du
Integration by Decomposition
Reverse Chain Rule
This method of integration is used when the integration is of the form ∫g'(f(x)) f'(x) dx. In this case, the integral is given by,
∫g'(f(x)) f'(x) dx = g(f(x)) + C